### BE2M31CZS / BE2M31DSPA exercise - Discrete-time systems, difference equation, transfer function.

• Properties of 2nd order systems, inverse Z-transform of higher-order systems.

• For discrete-time systems described by following difference equations draw block scheme, compute difference equation, transfer function, zeros and poles of transfer function, impulse response, and estimate frequency response.

1. y[n] = x[n] + 2 x[n-1] + 1.5 y[n-1] - y[n-2]

2. y[n] = 0.4455 x[n-1] + 1.2728 y[n-1] - 0.81 y[n-2]

3. y[n] = x[n] + 0.81 x[n-2]

4. Observe the change of the frequency response and zero-poles locations for the following systems:
• y[n] = x[n] + 0.81 x[n-2]
• y[n] = x[n] + 0.64 x[n-2]
• y[n] = x[n] + 0.925 x[n-2]
• y[n] = x[n] - 0.81 y[n-2]
• y[n] = x[n] - 0.64 y[n-2]
• y[n] = x[n] - 0.925 y[n-2]

5. Only in MATLAB: "all=pole" filter with the pair of complex conjugate poles p_1 = 0.9 *exp ( +/- j * pi/4 )

6. 7. y[n] = x[n] + x[n-1] + x[n-2] + ... + x[n-9] ;

8. y[n] = x[n] - x[n-6]

9. y[n] = x[n] - x[n-9]

• For above described system observe in MATLAB:
• zeros and poles of transfer function (fnc zplane, roots)
• frequency response of the system (fnc freqz)
• impulse response of given system (fnc impz)
• check also interactive tool fvtool

Perform the filtering of white noise using the system No. 4. The noise should have uniform distribution and the length of 10000 samples. Display:

• 1st checked result (1 point):
• time waveforms of input and output signals,
• spectrograms of input and output signals computed with the length of short-time frame of 256 samples,
• frequency response of used filter,

• 2nd checked result (1 point):
• short-time power spectra of input and output signals computed from the short-time frame with the length of 256 samples,
• twoside smoothed estimation of PSD for input and output signals, the short-time frame should have the length of 256 samples,